Abstract
In this paper, we extend Hardy’s type inequalities to convex functions of higher order. Upper bounds for the generalized Hardy’s inequality are given with some applications.
Highlights
Faculty of Textile Technology, University of Zagreb, Prilaz Baruna Filipovica 28a, 10000 Zagreb, Croatia; Faculty of Civil Engineering, University of Zagreb, Fra Andrije Kačića-Miošića 26, 10000 Zagreb, Croatia
In this paper, we extend Hardy’s type inequalities to convex functions of higher order
Ω1 and obtain new inequalities that hold for n-convex functions
Summary
Ω1 and obtain new inequalities that hold for n-convex functions. The. Abel–Gontscharoff interpolation for two points and the remainder in the integral form is given in the following theorem (for more details see [11]). Suppose that all assumptions of Theorem 1 hold and let n, m ∈ N, n ≥ 2,. If (10) holds, φ(s) (α) ≥ 0 for s = 0, ..., m and (−1)r−s φ(m+1+r) ( β) ≥ 0 for s = 0, ..., r and r = 0, ..., n − m − 2 the right hand side of (10) is non-negative, i.e., the inequality (4). The following estimations for Hardy’s difference is given, under special conditions in Theorem 6 and Remark 3. We have the inequality ( x − α)( β − x )[h0 ( x )]2 dx
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